Numerical Derivative Using the Piecewise Uniform Mesh

In this paper, a technique of a piecewise-uniform meshes formed on an improvement finite difference algorithm for finding derivatives of functions. The purpose was to overcome difficulties which face numerical derivatives of functions with stiff formula, the main idea is that the formula includes some terms that can lead to rapid variation in the graph of the functions, which have recently been named singular layers in numerical analysis. The fundamental numerical difficulty is related to non-physical oscillations of the solution (instability) when the formula of the function dominates over the formula of its derivatives, this is a characteristic of many fluid flow problems. The use of Shishkin mesh to find derivatives of arbitrary degree and order is the novelty of this paper. The method was applied to find derivatives of some examples until third order and the results were compared with a previous study, mentioned in the paper, to derive the functions numerically.


Introduction
Functions are known to be Stiff ( ) , Its derivatives reach a certain order of that depends on the softness of the data can be limited by: Where and indicates a general positive constant independent of and the number of mesh points used. Kellogg and Tsan proved these estimates [1].
Recently, Shishkin showed that the regular and singular components of can be separated: have a representation of , where This Shishkin decomposition has played a major role in finite difference analysis and finite component methods on the meshes of Shishkin and other adaptive layer meshes in recent years. It was generally thought that decomposition (2) was necessary to demonstrate a uniform convergence of standard numerical methods on the meshes of the adapted layer. It should be noted that (1) and (2) are equivalent [2]. Prandtl originally introduced the term boundary layer in 1904 [3]. It is often easier to locate the boundary layer, it will be found analytically or by plotting the graph of the function, to illustrate this we consider a typical boundary layer function obviously this boundary layer is the one that fall at the boundary layer as in Prob.4- Fig. 2. We define the arithmetic representation of the boundary layer as follows: because of the function of the boundary layer ( ) and the separate measurement function ( ) so that ( ) as then the arithmetic width of the boundary layer corresponding to the boundary function ( ) for ( ) is the smallest value that we have meshes are a uniform mesh ̅ * + with spaced grid points for all .
But with this mesh none of the mesh points will be inside the boundary layer, unless if (i.e. when layer width ). Piecewise is a unified mesh installed on the boundary layer, consisting of two uniform meshes: a denser fine mesh is for the boundary layer and a coarse mesh is outside the boundary layer. The location of the shift point between the fine and the coarse mesh is the function of the stiffness parameter and a parameter of the discretization . The correct distribution of the grid points is to obtain an equal or comparable number of mesh points in the fine and the coarse meshes [4]. A numerical comparison between Shishkin Mesh (S-Mesh) and classic Uniform Mesh (U-Mesh) will be presented, on same certain finite difference operator, in section (8).

Maximum Absolute Value Norm
The norm comes with boundary layer functions that do not involve averages namely or the maximum norm defined by ‖ ‖ and it also follows this arithmetic law: The maximum absolute value norm is the only appropriate criterion for studying the phenomena of the boundary layer; this is because differences between distinct functions are detected, no matter how much [6]. The choice of the maximum error measurement rule is due to the need to measure the error in the very small ranges in which the boundary or interior layers occur. Other parameters, such as the root average square, include error means, which slow down rapid changes in solutions and thus fail to capture the local behavior of error in these layers. To be sure, these assertions are valid, for example, the function of the boundary layer and √ on ̅ , where ( ) Also known as -Norm, uniform norm, max norm, or infinity norm defined as maximum absolute values of its components [6]:

Shishkin Mesh
In particular, this area of numerical analysis has been strengthened by the contributions of Russian mathematician Grigorii Ivanovich Shishkin. We will describe the construction of Shishkin's formations and analyze the Shishkin solution, and we also aim to highlight the mainstream approach of Shishkin, which is evident from the wide range of problems that Shishkin has applied his methodology. The Shishkin mesh is a uniform mesh of piecewise.
What distinguishes the Shishkin network from any other uniform piecewise mesh is the selection of so-called mesh transition parameter(s), which is the point at which the mesh scale changes abruptly [7].

The modest composition of one dimension piecewise uniform mesh
The resulting piecewise uniform meshes blow, depends only on one parameter . We will denote, the Piecewise uniform meshes with mesh elements and one parameter , by ̅ :

1-A single Boundary Layer Mesh:
A simple example of a piecewise uniform mesh is constructed on the interval ( ), as follows; Select a point satisfactory and assume that , for some .
The point τ is called a mesh transition point which divides into two subintervals the fine one is ( ) and the coarse one is ( ) if the boundary layer is close to zero but the coarse one is ( ) and the fine one is ( ) if the boundary layer close to one. The

2-Couple Boundary Layers Meshes:
The following piecewise uniform fitted mesh is constructed if the boundary layer located at both boundary points and , the fine mesh must be in the neighborhood of each of these points. Thus, two transition points exit and three subintervals are needed. The simplest construct is to choose achieving , and determine the mesh transition points at and . Assuming that , with , the intervals ( ) ( ) and ( ) each is divided into equal mesh elements and it becomes uniform when , as in Fig. 1 (c).

3-Interior Layer:
As well as in the case of the boundary layer settled at the midpoint of the interval

On the accuracy of the proposed difference schemes (HMG)
The difference schemes we use are algorithm for numerical differentiation presented by The values of the function at all other nodes can be expressed as Taylor series in terms of the reference point as given in equation (2): Substituting into Eq. (7) we get This can be rearranged to take the following form: [4], [6], and [11].

Numerical results and Conclusion
In  Table 1 was provided for describe each of the test problems, details of the data, the size of the solutions, and to illustrate the figures and other tables.            The numerical results indicate that the new technique has an improvement about (86.5038105%) in Maximum error of the s-mesh method against the uniform mesh method [10], as in Table 6.